State-Space Realization of Periodic i/o Behaviors

Aleixo, José; Rocha, Paula

doi:10.1109/tac.2020.3008391

Cited by 6 publications

(12 citation statements)

References 15 publications

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“…Definition 2 ( 14 ). Let Σ = (𝐹 , 𝐺, 𝐻, 𝐽 ) be a linear time-invariant 𝑛-dimensional state-space system with 𝑚P inputs and 𝑝P outputs, for a given positive integer P. Define an 𝑛-chain of size 𝑠 generated by Σ as a sequence of matrices 1 𝑀 , … , 𝑠 𝑀 , each one of rank less than or equal to 𝑛, such that:…”

Section: Periodic State-space Representationsmentioning

confidence: 99%

“…Theorem 1 ( 14 ). Let Σ = (𝐹 , 𝐺, 𝐻, 𝐽 ) be a linear time-invariant 𝑛-dimensional state-space system with 𝑚P inputs and 𝑝P outputs, for a given positive integer P. Then, Σ is induced by a P-periodic state-space system of dimension 𝑛 if, and only if:…”

Section: Periodic State-space Representationsmentioning

confidence: 99%

“…It has been shown in 14 , that a linear time-invariant system Σ = (𝐹 , 𝐺, 𝐻, 𝐽 ) that generates an 𝑛-chain of size P − 1 is induced by a P-periodic state-space system Σ(𝑘) = (𝐴(𝑘) , 𝐵(𝑘) , 𝐶(𝑘) , 𝐷(𝑘)) such that:…”

Section: Periodic State-space Representationsmentioning

confidence: 99%

“…The minimal realization problem for linear time-varying systems is a fundamental topic in linear systems theory that has received several contributions as, e.g., 1,2,3,4,5,6,7,8 , in the last decades. More recently this issue has been addressed in the context of the behavioral approach where a strategy was proposed in order to achieve a state-space realization for periodic MIMO behaviors, see 14 . A first step towards the questions this work addresses was initially treated in 13 , where we have analyzed the issue of the state-space representation of periodic SISO behaviors only for the particular case of period equal to two.…”

Section: Introductionmentioning

confidence: 99%

“…A first step towards the questions this work addresses was initially treated in 13 , where we have analyzed the issue of the state-space representation of periodic SISO behaviors only for the particular case of period equal to two. Taking this last two works into account, as a starting point, we present some further results on the matrix chain used in 14 to produce such realizations and provide an algorithm that allows to obtain low-dimensional realizations for periodic MIMO behaviors.…”

Section: Introductionmentioning

confidence: 99%

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CMMSE: A low-dimensional realization algorithm for periodic input/output behavioral systems

Aleixo

Rocha

2022

Preprint

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The state-space realization of linear systems is of utmost importance in linear systems theory. After the realization problem for the time-invariant case has been solved, particular attention was paid to the case of linear periodic systems (see, e.g. 1,2,3,4,5,6,7,8). Recently, such systems have regained importance, for instance, in the context of coding theory (see9), where periodic convolutional encoders play an important role, 10. The majority of the contributions within this area concern the realization of transfer functions as well as impulse responses, thus excluding the case of input/output linear systems without coprime representations. By the end of the eighties of the last century, Jan C. Willems (see11,12) suggested an approach (nowadays known as the behavioral approach) that considers a wider class of systems and allows to overcome this drawback. According to this approach, the central object in a system is its behavior which consists of all the signals that satisfy the system laws (also called system trajectories). Consequently, the behavior of a system with an input/output representation that is not coprime, contains more trajectories than the set of input/output signals defined by the system transfer function. Our work takes this fact into account. Based on results already obtained in 13,14, we revisit the problem of the realization of linear periodic MIMO behaviors and give further insight into this problem, which allows setting up an algorithm to compute a low-dimensional state-space realization of a periodic behavior. The proposed algorithm is based on a chain decomposition of suitable matrices.

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Section: Periodic State-space Representationsmentioning

confidence: 99%

Section: Periodic State-space Representationsmentioning

confidence: 99%

Section: Periodic State-space Representationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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CMMSE: A low-dimensional realization algorithm for periodic input/output behavioral systems

Aleixo

Rocha

2022

Preprint

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Computational methods for periodic systems: Recent developments

Aleixo,

Serôdio

2024

Math Methods in App Sciences

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In the past few decades, there has been a significant focus on computational methods for periodic systems. In some of these works, various considerations were outlined to establish and categorize algorithms for designing periodic discrete‐time state‐space systems. In a recent work, a direct algorithm was proposed to obtain low‐dimensional realizations from a proper lifted system in state‐space form. This algorithm, which lies in a category often called “fast” algorithms, takes advantage on the structure of lifted representations being highly efficient. In a 1993 work by Ching‐An Lin and Chwan‐Wen King, another direct algorithm was proposed that appears to be quite similar to the algorithm outlined recently. These two algorithms generate state‐space periodic realizations that are both uniform, meaning that the dimension of the state remains constant at each time instant. The algorithms proposed in these two works require both a rank test applied to two different types of matrices, and , respectively (in chronologic order). Although the former algorithm establishes an upper bound for the state dimension à priori, by determining the maximum rank of the matrices , the latter obtains an upper bound à posteriori, based on the rank of matrices . In this work, we proved that these matrices and have the same rank for all , and we use this information to revise the latter algorithm and assess whether it represents an improvement. The algorithm presented as well its modified versions that were coded using the Python programming language. Surprisingly, the revised algorithm did not demonstrate any significant improvement in terms of computational efficiency.

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State‐space realization of periodic input/output quaternionic linear systems

Aleixo,

Serôdio

2024

Math Methods in App Sciences

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The investigation of linear periodic systems is a prominent subject of research within the domain of linear systems theory. In this context, the state‐space realization of such systems is of particular significance. In fact, after successfully treating the realization problem for time‐invariant systems, various researchers have directed their attention toward investigating the case of linear periodic systems. Meanwhile, in the late 1980s, Jan C. Willems proposed an approach that broadened the range of systems studied, now referred to as the behavioral approach. This approach views the behavior of a system as its fundamental element, including all signals that adhere to the system laws (also called system trajectories). More recently, the behavioral framework has also been extended separately to periodic and quaternionic behavioral systems. Our work consists of considering linear periodic input/output quaternionic behavioral systems and using recent developments in order to obtain quasi‐minimal and uniform state‐space realizations.

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State-Space Realization of Periodic i/o Behaviors

Cited by 6 publications

References 15 publications

CMMSE: A low-dimensional realization algorithm for periodic input/output behavioral systems

CMMSE: A low-dimensional realization algorithm for periodic input/output behavioral systems

Computational methods for periodic systems: Recent developments

State‐space realization of periodic input/output quaternionic linear systems

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